Add diagview to obtain a view along a diagonal

[jishnub]

A function to obtain a view of a diagonal of a matrix is useful, and this is clearly being used widely within LinearAlgebra.

The implementation here iterates according to the IndexStyle of the array:

julia> using LinearAlgebra

julia> A = reshape(1:9, 3, 3)
3×3 reshape(::UnitRange{Int64}, 3, 3) with eltype Int64:
 1  4  7
 2  5  8
 3  6  9

julia> diagview(A,1)
2-element view(::UnitRange{Int64}, 4:4:8) with eltype Int64:
 4
 8

julia> T = Tridiagonal(1:3, 3:6, 4:6)
4×4 Tridiagonal{Int64, UnitRange{Int64}}:
 3  4  ⋅  ⋅
 1  4  5  ⋅
 ⋅  2  5  6
 ⋅  ⋅  3  6

julia> diagview(T,1)
3-element view(::Tridiagonal{Int64, UnitRange{Int64}}, StepRangeLen(CartesianIndex(1, 2), CartesianIndex(1, 1), 3)) with eltype Int64:
 4
 5
 6

Closes #30250

[mcabbott]

Seems OK, it's used a lot internally, and this is usually the only reason to want diagind.

However, it seems like the one-arg form should understand structured matrices, so that e.g. diagview(T) isa UnitRange. And likewise diagview(UpperTriangular(rand(4,4))) should probably act on the parent (which has linear indexing).

What diagview(UpperTriangular(rand(4,4)), 2) should do is less obvious... it could make an efficient view(::Matrix) in the useful case, but not for the -2 case, hence at the cost of type instability.

[jishnub]

Type-stability is indeed the reason I didn't specialize this for banded matrices. This parallels how diag doesn't copy the bands, but writes to a similar vector. While the single-argument method may be specialized, I wonder if this is particularly useful? Accessing any band other than the principal diagonal would require the 2-argument method. We would also want to ensure that diagview(A) and diagview(A,0) return the same type, so whether we specialize the one-argument method goes along with whether we want type-stability in the two-argument method.

[mcabbott]

We would also want to ensure that diagview(A) and diagview(A,0) return the same type

Why do we want this?

[jishnub]

Mainly for ease of use. Having code that works for diagview(A) but not for diagview(A,0) doesn't sound ideal.

[mcabbott]

I think you're suggesting someone may pass diagview(A) to a function which accepts only ::Vector, and may be surprised if they later change to diagview(A, 0). That seems unlikely to me, and sort-of backwards? If you ask for a view of some unknown AbstractMatrix, you generically expect some AbstractVector whose type is complicated... and if you accept that, then when someone passes in A::Diagonal, getting a simple Vector seems like it should be fine.